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Bsc third year Newtown rephson method
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The Newton-Raphson method is an iterative numerical technique used to find approximate solutions to equations of the form
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erC.bc..kl n!h
Hence, f() - fa)+Afa)+ x-a (*-a-),?f(a)+...
2!
h h h
Putting x-a=mh, we have
f(ut mh) =f(a)+mAf(a)+ m
(m-)A² f(a)+...
2!
28. Algorithm for Difference Interpolation Formula
. Input the Newton's
D
of xfor values of x,, and corresponding values off(x,). Also input the values
Stepwhi2.ch S) to be calculated.
is
Prepare divided difference table by using
, Numerical Methods d
16 | | Unified
f(x)- A
X...-1
values of divided differences in the formula
Step 3.
Substitute the
i=|X...
reduces to
Remark: Newton's
the
divided
values of
difference formula
the argument are equidistant, Newton-Gregsry
ie..b-a=C-b=d..
tr
difference formula, if
-|-k= h, then
fO)- fa) f(a)
Af(a)
A"f(a)
A"f(a) = n!h"
etc.
bc-k
x-a(x- a-l) 1
Hence, f) = f(a) + A f(a) +
h h
Puttingx- a= mh, we have
f(at mh) = f()+ mAf(a) t m(m-)?
2!
f(a).
ILLUSTRATIVE EXAMPLES
given that f(0) = 8,f(l)=.
Example 1. Find the form of the functiony=f(r),
f(4) = 68,f(5) = 123. Also determinef(2). difference interpolation formula, for w
Solution:We use the Newton's divided
we need the following divided difference table:
Divided Difference Table
Af) Af)
11-8
=3
1-0
19-3
11 =4
1 4-0
68-11
=19 5-0
9-41
4-1
$5-19
4 68 =9
5-1
123- 68
=55
5-4
5
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